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Sigma DPMO Converter

Converts between defects per million opportunities, yield and sigma level, including the 1.5 sigma shift. Use it to express process performance on the Six Sigma scale.

Last updated
Decision Canvas

Calculator overview

Inputs and outputs

This summary comes from the calculator's published input and output contract.

Inputs

Defects Observed Conditional
About this input

The number of defects found, in count. Used with the units inspected and opportunities per unit to compute the defects per million opportunities.

Unit count Default 12 Range 0 to 5000 (conditional)
Defects Per Million Opportunities Conditional
About this input

The defect rate entered directly, in defects per million opportunities, when that is the known figure.

Unit ppm Default 6210 Range 0 to 1000000
Input Mode
About this input

Which quantity you are entering to start from: raw defect counts, a yield, a defects-per-million figure, a sigma level or a capability index. It selects which inputs the tool reads.

Default Defects, units and opportunities Allowed Defects, units and opportunities, DPMO, Yield percentage, Sigma level, short term, Cpk
Minimum Process Capability Index Conditional
About this input

Cpk, the capability index accounting for off-centring, as an index. When this is the chosen input mode the tool converts it to a sigma level, yield and defect rate.

Unit index Default 1
Opportunities Per Unit
About this input

The number of independent ways a single unit can be defective, in count. Counting opportunities generously lowers the defects-per-million figure, so keep the definition consistent when comparing processes.

Unit count Default 5 Range At least 1
Process Steps
About this input

The number of sequential steps in the process, in count. Used to combine each step's yield into a rolled throughput yield across the whole process.

Unit count Default 1 Range At least 1
Sigma Level Short Term Conditional
About this input

The short-term sigma level, which includes the 1.5 sigma shift. When this is the input mode the tool works back to a defect rate and yield from it.

Unit sigma Default 4
Units Inspected Conditional
About this input

The number of units examined, in count. It is the denominator, with opportunities per unit, for the defect rate.

Unit count Default 1000 Range At least 1
Yield Conditional
About this input

The proportion of opportunities with no defect, entered as a percent. When this is the input mode the tool derives the defects per million, sigma level and equivalent indices from it.

Unit % Default 99.379 Range 0 to 100

Outputs

Defect Proportion
About this output

The proportion of opportunities that are defective, as a fraction between 0 and 1. It is the defects per million divided by one million.

Unit fraction
Defects Per Million Opportunities Result
About this output

The computed defect rate, in defects per million opportunities, the common currency for comparing process quality.

Unit ppm
Defects Per Unit
About this output

The average number of defects per unit, counting all opportunities on the unit. It can exceed 1 when a unit carries several defects.

Unit dpu
Minimum Process Capability Index Result
About this output

The equivalent Cpk for the computed defect rate, as an index, read as the short-term sigma level divided by three. It treats all defects as falling in one tail, which is the conventional one-sided reading; a heavily two-tailed process will show a slightly generous number here.

Unit index
Model Status
About this output

The overall check on your entries, shown above the results. It reads OK when the inputs are usable, NOT VALID with a reason when an entry makes the model meaningless, or CHECK with a reason when a result is valid but worth a second look. Read it before you trust the numbers below.

No unit declared
Process Capability Index
About this output

The equivalent Cp a centred process would need to reach this sigma level, as an index. It is a translation of the sigma level, not a measured capability.

Unit index
Rolled Throughput Yield
About this output

The probability a unit passes every process step with no defect, as a fraction between 0 and 1, computed as the single-step yield raised to the number of steps. That assumes the steps are independent and share this yield; a chain with mixed step yields multiplies their individual values instead.

Unit fraction
Sigma Level Long Term
About this output

The long-term sigma level, without the 1.5 sigma shift, as reported directly from the observed yield or defect rate.

Unit sigma
Sigma Level Short Term Result
About this output

The computed short-term sigma level, including the customary 1.5 sigma shift, so a centred six-sigma process reads 6.0. Quote which convention you use: the long-term line below is the same performance without the shift, 1.5 lower. At a defect rate of exactly zero or one the level shows the model's cap rather than infinity.

Unit sigma
Yield Result
About this output

The computed yield, as a percent: the share of opportunities with no defect.

Unit %

What it is

The Sigma DPMO Converter moves between the several ways Six Sigma work expresses the same process performance. Start from whichever figure you have (raw defect counts, a defects-per-million rate, a yield percentage, a short-term sigma level, or a capability index) and it reports all of the others.

It also computes rolled throughput yield across a number of sequential process steps, and reports both the long-term sigma level read directly from the data and the short-term level that includes the customary 1.5 sigma shift.

Use it to translate between conventions and to compare processes on a common scale. Every conversion here rests on assumptions that are stated below and that do not always hold.

Methodology

Purpose and model boundary

This calculator converts among an observed defects/opportunities rate, DPMO, yield, a shifted short-term sigma level, and a one-sided Cpk interpretation. It also reports defects per unit and a simple rolled throughput yield. It is a mathematical converter, not a process-capability study: it does not test stability, normality, independence, measurement error, or whether the opportunity definition is appropriate.

Inputs and units

The selected input mode determines which quantity is authoritative:

  • Defects, units and opportunities: nonnegative integer defects, positive units inspected, and positive opportunities per unit;
  • DPMO: defects per one million opportunities, bounded from 0 to 1,000,000;
  • Yield percentage: 0% to 100%;
  • Sigma level, short term: the convention that includes a 1.5-sigma shift;
  • Cpk: a one-sided capability-index interpretation.

Process steps is a positive integer used only for rolled throughput yield. Opportunities per unit also converts the normalized defect proportion to defects per unit.

Governing relationships

The chosen mode is first normalized to defect proportion q:

  • counts mode: q = defects / (units * opportunities per unit);
  • DPMO mode: q = DPMO / 1,000,000;
  • yield mode: q = 1 - yield_fraction;
  • shifted short-term sigma mode: q = 1 - Phi(sigma_short - 1.5);
  • Cpk mode: q = 1 - Phi(3 * Cpk).

The workbook then computes

  • DPMO = 1,000,000 * q;
  • yield = 1 - q;
  • DPU = defects / units in counts mode, otherwise q * opportunities per unit;
  • RTY = yield ^ max(1, process steps);
  • sigma_long = Phi^-1(1 - q);
  • sigma_short = sigma_long + 1.5;
  • reported long-term one-sided Cpk equivalent = sigma_long / 3;
  • reported shifted capability equivalent = sigma_short / 3.

Phi is the standard-normal cumulative distribution. The inverse uses Acklam's rational approximation. At the endpoints the workbook caps the inverse-normal result at -9 or +9 instead of returning infinity.

Calculation sequence

  1. Map the selected mode to its code and read only the mode-specific visible input.
  2. Normalize that input to q using the relevant equation above.
  3. Derive DPMO, yield, DPU, long- and short-term sigma, and both index equivalents from the common q.
  4. Raise the single-step yield to the declared number of process steps for RTY.
  5. Apply the model-status precedence before presenting the conversion as usable.

Outputs and interpretation

DPMO and yield describe opportunities, not necessarily defective units. DPU can exceed one when a unit contains several defects. The sigma and Cpk results are one-sided normal equivalents derived from the entered rate; they are not fitted process parameters. The 1.5 shift is a reporting convention, so comparisons must identify whether the shifted or unshifted figure is being quoted.

Validation and status logic

The workbook evaluates status in this order:

Condition Returned status
Selected mode produces q < 0 NOT VALID: the selected mode gives a negative defect rate
Selected mode produces q >= 1 NOT VALID: the defect rate is 100% or more
Counts mode has units inspected <= 0 NOT VALID: units inspected must be greater than zero
q = 0 CHECK: zero defects observed; the true defect rate is uncertain at this sample size and the sigma level shown is the model's cap
sigma_long < 0 CHECK: long-term sigma is negative, meaning more than half the output is defective
None of the above OK

The published limits also enforce positive opportunities and process steps, integer count fields, valid yield/DPMO ranges, and mode-specific visibility before the workbook call.

Assumptions and limitations

The opportunity count is a user-defined denominator and must remain consistent across comparisons. The normal-equivalent conversion treats defects as one-tail probability; it is not equivalent to a demonstrated two-sided normal capability study. The 1.5-sigma shift is conventional rather than measured. RTY assumes independent sequential steps with the same per-step yield; a process with different step yields should multiply those individual yields instead. Results are point estimates and do not include confidence intervals, so zero observed defects does not prove a zero underlying rate.

Restrictions and non-computing states

The workbook does not estimate confidence intervals or support different yields by process step. Invalid unit, defect, opportunity, or step counts produce a non-computing state identified by the status message; zero observed defects remains a finite-sample observation rather than proof of zero underlying defect probability.

Errors and warnings

A rejected entry means a mode-specific value, type, or bound did not satisfy the published input rules. Workbook NOT VALID means the normalized defect proportion is outside its usable domain. Workbook CHECK retains the conversion but calls out a statistical boundary or very poor yield. A network, authorization, or calculation-service failure is a service condition, not a quality level.

References

The workbook derives its relations rather than reproducing any table, chart or figure from a specification, standard or agency publication. The conversions are normal-distribution tail lookups and elementary arithmetic; no sigma conversion table is reproduced.

The 1.5 sigma shift is a widely used convention rather than a statistical result, and the 3.4 defects per million figure follows from it rather than from a six-sigma tail. Both sigma levels are reported here so that a figure taken from this tool can be matched to whichever convention the source it is compared against uses.

Additional source notes migrated from Methodology

The workbook uses the standard DPMO identity, the standard-normal CDF, Acklam's inverse-normal rational approximation, and the explicitly stated 1.5-sigma convention. The page's References section contains the coefficient-source and licensing details.

Frequently asked questions

Why does the tool show two sigma levels, and which one should I quote?
Because there are two conventions in circulation and quoting the wrong one silently changes the answer by 1.5. The long-term level is read directly from the defect rate you supplied and is what the data say. The short-term level adds the customary 1.5 sigma shift and is what most Six Sigma reporting means by "sigma level". Quote whichever matches the convention of whatever you are comparing against, and say which one it is.
Where does the 1.5 sigma shift come from, and is it real?
It is a convention, not a measurement. A truly centred process with six standard deviations to the nearer specification limit would produce about two defects per billion, not 3.4 per million. The Six Sigma convention allows 1.5 standard deviations of long-term drift when translating a short-term capability study into expected production performance, and the familiar 3.4 figure is the tail at 4.5 sigma. Whether 1.5 describes your process is an empirical question this tool cannot answer.
Why does the opportunity count change my sigma level so much?
Because defects per million opportunities is a rate per opportunity, so the denominator is a choice. The same 12 defects in 1,000 units gives 2,400 per million at 5 opportunities per unit, and 12,000 per million at 1: a short-term sigma level of 4.32 against about 3.76, for identical product. Generous opportunity counting raises the sigma level for free. Defects per unit is reported alongside precisely because it does not have this weakness, and it is the better figure for comparing across organisations.
Why is my Cp exactly 0.5 higher than my Cpk?
Because both are derived from the sigma levels rather than measured from your process. `Cp` is the short-term level divided by three and `Cpk` is the long-term level divided by three, and the two levels differ by exactly the 1.5 shift. The gap is therefore always 0.5 and it tells you nothing about how off-centre your process actually is. To measure real centring you need the process data, not a defect rate.
Can I trust the rolled throughput yield for a long process?
Treat it as a first approximation. It computes yield raised to the number of steps, which assumes every step has the same yield and that failures at different steps are independent. Real processes have steps with different yields and correlated failures (one upstream batch problem causing several downstream rejections), and in both cases the true rolled yield is worse than this formula suggests. The tool does not accept per-step yields.
I found zero defects. Why does the tool not report an infinite sigma level?
Because zero defects in a finite sample does not mean a zero defect rate; it means the rate is below what your sample size could detect. The tool caps the reported sigma level and says so in the status rather than dividing by a rate of zero. To claim a very high sigma level you need a sample large enough to have plausibly seen a defect if one existed at that rate.
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This page is provided by LogicCommons for informational purposes only. Results are analysis outputs computed from the inputs you supply and are not engineering advice, a design, or a substitute for review by a licensed professional under the codes adopted where the work is built. Verify all inputs and results independently.

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